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Hyperreal number

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In mathematical logic, the hyperreal numbers or nonstandard reals (usually denoted as *R) is an ordered field which is a proper extension of the ordered field of real numbers R. An important property of *R is that it has infinitely large as well as infinitesimal numbers to R, where an infinitely large number is one that is larger than all numbers representable in the form

1 + 1 + ... + 1

The use of the definite article the in the phrase the hyperreal numbers is somewhat misleading in that there is not a unique ordered field that is referred to, in most treatments.

The study of these numbers, their functions and properties is called nonstandard analysis; some find it more intuitive than standard real analysis. When Isaac Newton and Gottfried Leibniz introduced differentials, they used infinitesimals and these were still regarded as useful by Leonhard Euler and Augustin Louis Cauchy. Nonetheless these concepts were from the beginning seen as suspect, notably by Bishop Berkeley, and when in the 1800s calculus was put on a firm footing through the development of the epsilon-delta definition of a limit by Cauchy, Karl Weierstrass and others, they were largely abandoned.

However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. Because his theory in its full-fledged form makes unrestricted use of classical logic and set theory and, in particular, of the axiom of choice, it is suspected to be nonconstructive from the outset. The construction given below is a simplified version of Robinson's more general construction and is generally attributed to E. Zakon.

1 References
2 External links

Contents

1 Properties

2 Construction

3 Infinitesimal and infinite numbers

Properties

The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology.

The hyperreals are defined in such a way that every true first-order logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers is also true if we presume that it quantifies over hyperreal numbers. For example, we can state that for every real number there is another number greater than it:

$\forall x \in \mathbb{R} \quad \exists y \in\mathbb{R}\quad x < y$

The same will then also hold for hyperreals:

$\forall x \in \star \mathbb{R} \quad \exists y \in\star \mathbb{R}\quad x < y$

Another example is the statement that if you add 1 to a number you get a bigger number:

$\forall x \in \mathbb{R} \quad x < x+1$

which will also hold for hyperreals:

$\forall x \in \star \mathbb{R} \quad x < x+1$

The correct general statement that formulates these equivalences is called the transfer principle. Note that in many formulas in analysis quantification is over higher order objects such as functions and sets which makes the transfer principle somewhat more subtle than the above examples suggest.

The transfer principle however doesn't mean that R and *R have identical behavior. For instance, in *R there exists an element w such that

$1<w, \quad 1+1<w, \quad 1+1+1<w, \quad 1+1+1+1<w, .\ldots$

but there is no such number in R. This is possible because the nonexistence of this number cannot be expressed as a first order statement of the above type. A hyperreal number like w is called infinitely large; the reciprocals of the infinitely large numbers are the infinitesimals.

Construction

We are going to construct the hyperreals via sequences of reals. In fact we can add and multiply sequences componentwise ; for example,

$(a_0, a_1, a_2, \ldots) + (b_0, b_1, b_2, \ldots) = (a_0 +b_0, a_1+b_1, a_2+b_2, \ldots)$

and analogously for multiplication. This turns the set of such sequences into a commutative ring A. Moreover, we can identify the real number r with the sequence (r, r, r, ...) and this identification preserves the corresponding algebraic operations of the reals.

We also need to be able to compare sequences. This is more delicate; indeed, we can define a relation between sequences in a componentwise fashion:

$(a_0, a_1, a_2, \ldots) \leq (b_0, b_1, b_2, \ldots) \iff a_0 \leq b_0 \wedge a_1 \leq b_1 \wedge a_2 \leq b_2 \ldots$

but here we run into trouble, since some entries of the first sequence may be bigger than the corresponding entries of the second sequence, and some others may be smaller. It follows that the relation defined in this way is a only a partial order. To get around this, we have to specify "which positions matter". Since there are infinitely many indices, we don't want finite sets of indices to matter. A consistent choice of "index sets that matter" is given by any free ultrafilter U on the natural numbers which does not contain any finite sets. Such a U exists by the axiom of choice. (In fact, there are many such U, but it turns out that it doesn't matter which one we take.) We think of U as singling out those sets of indices that "matter": We write (a₀, a₁, a₂, ...) ≤ (b₀, b₁, b₂, ...) if and only if the set of natural numbers { n : a_n ≤ b_n } is in U. This is a total preorder and it turns into a total order if we agree not to distinguish between two sequences a and b if a≤b and b≤a. With this identification, the ordered field *R of hyperreals is constructed. From an algebraic point of view, U allows us to define a corresponding maximal ideal I in the commutative ring A, and then to define *R as A/I; as the quotient of a commutative ring by a maximal ideal, *R is a field.

For more information about this method of construction, check out ultraproducts.

Infinitesimal and infinite numbers

A nonstandard real number r is called infinitesimal if it is smaller than every positive real number and bigger than every negative real number. Zero is an infinitesimal, but non-zero infinitesimals also exist: take for instance the class of the sequence (1, 1/2, 1/3, 1/4, 1/5, 1/6, ...) (this works because the ultrafilter U contains all index sets whose complement is finite).

A non-standard real number x is called finite (or limited by some authors) if there exists a natural number n such that -n < x < n; otherwise, x is called infinite. Infinite numbers exist; take for instance the class of the sequence (1, 2, 3, 4, 5, ...). A non-zero number x is infinite if and only if 1/x is infinitesimal.

The finite elements of F of *R form a local ring, with the unique maximal ideal S being the infinitesimals; the quotient F/S is isomorphic to the reals. Hence we have a homomorphic mapping, st(x), from F to R whose kernel conisists of the infinitesimals which sends every element x of F to a unique real number whose difference from x is in S; which is to say, is infinitesimal. Put another way, every finite nonstandard real number is "very close" to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x. This operation is an order-preserving homomorphism and hence well-behaved both algebraically and order theoretically: